A classification of regular -balanced Cayley maps on dicyclic groups

A Cayley map CM ( G , S , p ) is a 2-cell embedding of a Cayley graph Cay ( G , S ) into an orientable surface such that all vertex-rotations correspond to the cyclic permutation p of the generating set S . It is called regular if its automorphism group acts regularly on the dart set. A regular Cayley map M = CM ( G , S , p ) is called t -balanced if p ( x ) − 1 = p t ( x − 1 ) for every x ∈ S . In this paper, we classify the regular t -balanced Cayley maps on dicyclic groups for all t . As a result, all such maps are 1-balanced.